Bypassing Gödel

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BYPASSING GÖDEL

Floyd and Putnam on Wittgenstein’s Remarks, or the dispensability of mathematical truth

Abstract:

Following the debate launched by Floyd and Putnam, it is proposed that, after the demonstration of Gödel, stand pending issues that are not purely philosophical, but regarding how to judge the correctness of the interpretations (Yablo’s paradox is proposed for the heuristic comparison, instead of that of Richard or Epimenides), the mathematical content, and the referential ability (the Frege’s “sense”) of the undecidable sentence.

Previously we try to rebuild a formulation of it, intending to avoid both the ambiguities of the informal versions, as the cryptic aura that surround the gödelian original.

In closing, it is proposed to investigate whether the undecidable sentences form a recursive set. If yes, they can allow to the automation of deductive systems of numeracy, avoiding the truth issues. The core thesis is that the self application of demonstrability is homologous to the division by zero: indefinable and not necessary to the complete sense of arithmetic.

As a post scriptum, the reconstruction of formalism attempted by Gödel is remembered: he was the first in attempt to bypass his own negative results.

[ This is a tentative translation of this Spanish draft:

Eludiendo a Gödel.doc

Please feel free to suggest any changes in the translation or style. ]

Although it is not uncommon in philosophy that a debate occupies the

attention for centuries, yes it is that, having agreed 85 years ago about the correctness of a mathematical proof, still it is discussed not only the philosophical meaning of it, but also about what is it has demonstrated. Perhaps a similar precedent can be found in the discovery of non-Euclidean geometries (still is discussed in some circles if entail a refutation of Kant’s theories, and if the physical space is Euclidean or not).

This is the case referred to in the Appendix III of the Wittgenstein’s

“Remarks on the foundations of mathematics”, about the first Gödel’s incompleteness theorem, which have been the focus of the papers of Floyd &


Putnam [2000], Steiner [2001], Rodych [2003], Bays [2004], Berto [2009] et alter. Bernays [1959] and Shanker [1988] also were precedents.

This is not a purely historical question, about “what Wittgenstein meant”, or

philosophical, like “what means the fact that Gödel proved”, but something more essential: “what had he proved?”. For a statement that claims the non-deductibility of its own “numerical image” it seems to lack reference. The logic mathematical or meta mathematical academics, tried from the outset to terminate any discussion with the verification of the formal correctness of the Gödel's proof, that on the other hand no one (nor Wittgenstein) has been questioned. What that gives us this proof, however, it is not something that can end the debate, but only restart it from other foundations.

Some philosophers (from Bertrand Russell1, to John R. Searle [1997], § 2,

for instance) have ventured that, although formally correct, the undecidable sentence lacks “substance” because it is a long logical-mathematical surrounding to reiterate the old semantic problem of self-reference. But the problem with this type of objection is that formal systems were designed just for start empty and lacking in substance, to be interpreted and then, once the formal correction is established, give them substance, i.e., reference. And it is very difficult to prove that there is no possible interpretation; for a syntactically correct sentence, it is always possible to find some interpretation, even in the field of language itself. The Gödel’s proof opened at least as many unknowns than those that have closed. We will list some of them that remain opened.

Previously, we will try a kind of philological reconstruction of the form of

represent the undecidable statement, because, from 1934, in which Gödel himself showed at the Institute for Advanced Study in Princeton a proof less technical than that of 1931 (notes were taken by Kleene and Rosser and mimeographed at Princeton University, and then reproduced in Davis [1965]), the versions more or less free and informal were the standard. In Ladrière [1957] there is a sort of historical record of versions of the theorem).

What we will try in the following paragraphs is to show the version of the

undecidable statement that we will used; the reader may find it obvious, but we detail only to lay the foundations for a consensus.

1 “If you can spare the time, I should like to know, roughly, how, in your opinion, ordinary mathematics –or, indeed, any deductive system– is affected by Gödel’s work” Letter to Leon Henkin, 1 April 1963, quoted by Shanker[1988], IV, p. 66.


The well known and central claim of the 1931 article is the proposition VI:

For every ω-consistent recursive2 class k of FORMULAS there are recursive CLASS SIGNS r such that neither v Gen r nor Neg(v Gen r ) belongs to Flg(k) (where v is the FREE VARIABLE of r). As cryptic as this may seem, it says only that for each system like Peano-

Russell, there is a parallel class of numeric code of formulas, in which there is a specific one, r, such that neither its generalization nor the generalization of its negation are deductible from the axioms.

It is generally accepted that the undecidable sentence spoken of the previous

proposition is obtained from that showed in the red square in the image, taken from the original in German:

2 What Gödel called "recursive class" is called today "primitive recursive class", because the former term is understood today to mean "general recursive", whose recognition algorithm can be of infinite duration. Instead, primitive recursive classes' recognition algorithms necessary end.


That is: 19 (x) { x ¬Bx [ Sb ( p ) ] } Z(p) That means: “For all x: x is not a number of a proof (B=“Beweis”) of the

substitution in p (that is the code number of an open formula) the code number of the variable ‘y’ (which is 219) by the code of p (itself).”

In the final note 1 we will explain in detail the symbolism, and the

mechanism that produces the “fixed point”, syntactic analogous of the semantic phenomenon of self-reference.


It has also general acceptance that, if there is a demonstration of this formula, we could construct a proof of its negation. Therefore, if the system is ω-consistent the formula is undecidable3.

Beyond this consensus, the following metamathematical issues are open: 1st) If the general argument of Proposition VI can be interpreted or back-

translated in the natural English as some paraphrase of “if P is the sentence ‘P is not provable’, then P is true”.

2nd) What is the proper mathematical content (if there is one), that is, what

is the simple fact, and the numbers, properties and relationships, that Principia Mathematica “and related systems” cannot prove, and what is the status of these entities in the not formalized Theory of Numbers .

3rd) If may be the case that it were syntactically correct but lacking of

coherent interpretation, as the example from Chomsky[1957]: “Colorless green ideas sleep furiously”. More technically, if it is true in some model.

We will outline the answers that might be proposed. 1st) Interpretation in a natural language. Wittgenstein’s interpretation in informal German (here in English) language

was suggested by Gödel himself at his introduction (part 1 of the original paper), but for illustrative purposes only [“In spite of appearances, there is no faulty circularity about such a proposition, since it begins by asserting the unprovability of a wholly determinate formula (namely the q-th in the alphabetical arrangement with a definite substitution), and only subsequently (and in some way by accident) does it emerge that this formula is precisely that by which the proposition was itself expressed.” In footnote 15, p. 175 original, p. 598 van Heijenoort]. Based exclusively on syntax, although the analysis of operations conducts to a close interpretation sought as the self-reference that appears in the introduction (or something we take as self-reference), the mechanism is complex enough to avoid

3 ω-consistent means that there is consistency in terms of quantifiers, that is,

if it holds that applies a formula for any value of x, then does not follow the quantified formula “there is an x” that does not meet the formula. “Undecidable” is a sentence that is not provable nor refutable in a formal system.


obvious circularity. It is, some as (and using the word chosen by Gödel himself in note 48a) a “transfinite” self-reference4.

Although Gödel himself favored comparison with the Richard’s paradox and

that of Epimenides, (“the liar”, see final paragraph of p. 175 original, p. 598 van Heijenoort), it is reasonable to argue that, if he had known the Yablo paradox, he would have preferred it above all, because it is not circular, self reference free, and of infinitary nature5.

We try to substantiate this. If we are to find a sentence in natural language (in

our case, English without technical or theoretical definitions) that corresponds through a back-translation as rigorous as possible, with the undecidable statement (UND hereafter) without numerical quotations (numeric codes of formulas), we could follow this procedure: transcribe the non-numeric symbols to natural language, and when we find a numerical quotation, subordinate the rest with the conjunction “that”. Then we need a three step back-translation: a) transcribe the numerical quotations to the simbolic formulas’ language; b) if there is a substitution function, resolve it; c) translate the simbolic formulas to natural language. If there is another numerical quotation at this point, we restart the whole process with the partial result as new argument. If everything is in natural language, we completed. But instead, we would obtain an infinite sequence like this:

It is not provable that It is not provable that It is not provable that It is not provable that .... It is not provable (x) ¬Proof(x, Subst( #Fmlº, num(#Fmlº), 219 )

) Here we meet Yablo’s paradox. His paradox was defined as an infinite list of

the form:

4 In the terminology of our days, this parallel between symbolic notation, and the numeric codes, is called "fixed point". A fixed point, in mathematics, is one that meets the following property for some function F: F(k) = k. In the present case, there is a slight variant from the *exact* fixed point, avoiding circularity: if t is the numerical code of an open formula Fml (x) with the free variable 'x', and 17 is the code of 'x', then Subst (t, num (t), 2 17) = num(t); see Boolos [2007], Lemma 17.1.

5 By the way, did Yablo obtained his paradox, thinking about how Gödel transformed the liar?


(S1) for all k >1, Sk is untrue (S2) for all k >2, Sk is untrue (S3) for all k >3, Sk is untrue ... “Suppose for contradiction that some Sn is true. Given what Sn says, for all k>n, Sk is

untrue. Therefore (a) Sn+1 is untrue, and (b) for all k>n+1, Sk is untrue. By (b), what Sn+1 says is in fact the case, whence contrary to (a) Sn+1 is true! So every sentence Sn in the sequence is untrue. But then the sentences subsequent to any given Sn are all untrue, whence Sn is true after all! I conclude that self-reference is neither necessary nor sufficient for Liar-like paradox.” (Yablo [1993]).

If we interpret each Sn like pointing to the nth line of the development of

auto-substitution, rather than a reference to other sentences of the same list, such as pointing to the nth line of development of the autosubstitution in the undecidable statement, translated into natural language, and replacing “not true” by “not provable”, and since each one is provable if and only if they are not provable the following in the list, the result is “as if” they assert their own unprovability.

Bays[2004] wrote: “First, I concede a point: there is nothing in the formal

structure of P —i.e., in P’s very syntax— which forces us to interpret P as ‘P is not provable.’ Nor, as Floyd and Putnam notice, does Gödel’s original proof require such an interpretation. So, if these were the only ways of providing a mathematically significant interpretation of P, then Floyd and Putnam would be right in challenging the claim that P means ‘P is not provable’.”

However, the interpretation of UND as generating a implied endless

succession of formulas of the type: Pi = “Pi+1 is unprovable” does seem to be strongly supported by the syntax of the formula. The need of such interpretation for the proof of the theorem, follows by the construction of each Pi+1, that is a numeric transliteration from Pi, so the later is provable if and only if the former is provable, and then, is asserting implicitly its own unprovability.

2nd) What is the “mathematical fact”? The mathematical construction is impressive; the undecidable statement is

built with hundreds of elementary functions and logical combinations, based on the 46 definitions in pp. 182-186 of the original. With the techniques used by Gödel theorem were born the recursion theory and that of computability.


But finally, the only mathematical fact that proves is that if there is a numerical sequence that satisfies the relation Bew with the numeric code of UND, there is a sequence of back-translatable formulas of the numeric codes, to prove his denial; and vice-versa (concluding in the ω-inconsistency).

But is this an arithmetic fact? Actually, the arithmetic seem to be only a mean

to demonstrate the incompleteness of formal language. There are many versions of the same theorem using different numbers and different symbols. Any theorem can have many versions, and many proofs, of course, but there is a content which remains unchanged. And in this case also, only that the numbers just are not the invariable content. It is the formal structure.

An indication of this is that the famous expression “17 Gen r”, the code

number of the undecidable statement, actually says nothing about the number 17 or other numbers that make up the “Relationszeichen” r (relation sign r), but uses them as mere tokens. In fact, the system of symbols can be changed without affecting the proof. Primitive logical connectives can be a single one (so-called “Sheffer strike”, or “not both”, negation of the conjunction, used by Quine [1951]), two (negation and implication, Mendelson [1964]) or three (negation, disjunction and universal quantifier, Gödel [1931]). Then the number of the variable ‘x’ in Godel is 217, in Mendelson is 213. So the expression that should be used with Mendelson’s numerical tokens is “13 Gen r”. But more surprising is that the second code number corresponds, when using Gödel’s tokens, to an unrecognized, malformed chain of symbols, and thus easy to prove that is a non-system chain, and thus, underivable; also its negation. Of course, the validity of the theorem is not affected even remotely. But it is clear that the specific numbers are not the target of the proof.

Returning to the analogy with the emergence of non-Euclidean geometries in

the nineteenth century, remember that the discovery that there was more than one possible geometry produced a great commotion among mathematicians and philosophers; but then it became clear that were applicable to surfaces (hyperbolic, elliptic, riemannian) others than plane. Nothing in these new geometries refutes the known geometry of the plane, nor adds nothing to it. Yes it was new the question about whether the physical space was Euclidean or not, i.e., whether physical applications oblige to exclude some of the possible geometries. Similarly, Gödel introduces a numerical coding parallel to the symbolic theory, transforming some code in a blind spot, that the formal system cannot discern if it's a theorem or is so its negation. But fails to establish something analogous to a “non-Euclidean” arithmetic. First, because the blind spot can be seen from another encoding, or other system, formal or intuitive, and secondly because it is a hiatus of its own


syntax, which can be bypassed, not with new axioms, but with deeper syntactic rules, regarding for functional singularities of self application.

The mysterious case of the number 17 could be an alternate explain for “the

Wittgenstein’s puzzling remark that ‘what is called losing in chess may constitute winning in another game.” (See Floyd and Putnam[2000] )

But there’s more: perhaps from Gödel system of tokens we could state that

of Mendelson, and so prove that 13 Gen r is not in the class of deductions in the second.

We might venture then, since the existence of undecidable statements is a

general fact of any language with quotation capability, that the arithmetic is used only to describe facts that aren’t of its scope, as used in physics or genetics, or more appropriately, in formal linguistics (generative grammars, see Chomsky[1957]). We could illustrate the law of the excluded middle with arithmetic: “2 + 1 = 3 OR 2 + 1 ≠ 3”; so it is not a proper arithmetic fact, but a logical one. The whole theorem, we guess, illustrated mathematically a problem of syntactic redundancy.

And one more question arises: if something as basic as the “protosintax”

(which is only logic with identity applied to strings of characters, see Quine [1940], § 59), is incomplete, then --is actually the number theory the cause of the incompleteness of its formal system? Some scholars reasoned that, if pure predicate calculus is complete, and the association with Peano’s arithmetic was incomplete, the cause was in the Peano’s axioms (in general, in the proper axioms of each incomplete theory).

But as there is no “pure” theories, independent of the logic of quantification,

it is not possible to check if proper axioms are the cause of incompleteness. It might be possible to investigate whether a theory is incomplete with the Peano’s axioms and the theory of identity as object language, leaving the whole apparatus of logic of quantification as a metatheory.

Worth a simile of optics to try to understand the situation: if we take two

mirrors separately, none of them will present endless aberrations. If we put the one in front of the other, we see that both contain infinite replicas of the image of the other. What this suggests is that it is not sure that the problem is only in the proper axioms of the theories of order 1, but in the logic when it embed some potent theories.

In this area it’s all for to do.


*** 3rd) Is it true in some model? We have seen that at the time of self-substitution indicated at UND, a

numeric quotation is formed, which in turn has indicated a self-substitution. We haven’t a rule that allows us to perform the operation within a quotation, but even if there were one rule, the result would be a quotation that contains a quotation... that has indicated a self-substitution, and so on.

Since we cannot reduce subordinated quotations that contain an invocation

to the Subst() function, the statement is not part of the universe of pure numerical terms that are the last references of any model that is intended to be used to interpret it (what is often called Herbrand interpretation).

In other words, for a formula has “sense” it must be possible to eliminate in

one of the members who compose the equations, all operators (also in the subordinate quotations). All formulas are necessarily logical combinations of equations, since the relationship of equality corresponds to the only predicate. Therefore, if a formula cannot be derived from another function that eliminates substitution in all the subordinate quotations, then this formula does not have a extensional or terminal sense. Any formula must be analyzed as a aggregates of logical combination of elementary formulas of the form:

Numeral1 + Numeral2 = Numeral3 Or: Numeral1 × Numeral2 = Numeral3 The aggregates may be infinite, but not the levels of nesting (there is the

difference between numerable and non numerable infinite, and recursive and non recursive aggregates).

If we accept this very basic criterion, the undecidable statement is neither

true nor false in the most immediate and elemental interpretation (Herbrand interpretation).

As to whether it is comparable to the syntactically correct but incoherent

sentences, as “Colorless green ideas sleep furiously”, there is a much tighter simile.


Consider the sentence “The present king of France is bald”. It is syntactically well formed, but undecidable because it makes no sense to say whether or not the king is bald; it is neither true nor false, at this level of language. But if we consider the subject as a definite description, it is equivalent to “There exists only one man who is the present king of France, and this man is bald.” Then it is simply false, because the first component of the conjuction is. (See Russell [1905]).

Ideally we would be able to demonstrate that there is no real code number of

formula from the substitution, because there is no final complete formula. But numerical quotation build by Subst() function exists, but in turn contains a call to itself with the same arguments. But as it is within a quotation, it is protected from all interpretation, so we cannot claim circularity. So it is syntactically correct, and it has a numerical reference, but that is the numerical quotation of a non resoluble substitution function.

There is another interpretation of syntactic operations of Gödel’s proof that

can be helpful. The open formula:

(S1) “x is unprovable”

we know that is not true nor false. But instantiating the free variable with the very quotation of the same

formula:

(S2) “ ‘x is unprovable’ is unprovable” the formula become provable (because if the open formula were provable, the generalization follow from it, so, no one formula would be provable), hence “true” in any interpretation.

Considering that “replacing the free variable ‘x’ in x for its own quotation” is

a functional description in terms of what we did manually to instantiate the variable, we have the same mechanism of the undecidable statement.

(S3) “The substitution of ‘x’ by its own quotation in x is unprovable.” (S4) “The substitution of ‘x’ in (S3) by its own quotation is

unprovable.” Should not (S4) to be provable if and only if so it is (S2)? The undecidability

slips through the crack in the ambiguity of what is a “closed formula” in a


quotation. While the quotation at (S2) has ostensibly a free variable, the quotation in (S4) has its own free variable (the same that is in S3) “camouflaged” as a function argument, and also quoted.

Consider also the Spanish translation of (S3): (S5) “La substitución de 'x' en x por su propia cita es indemostrable”. Let be (S6) similar to (S4), replacing (S3) by (S5). (S6) is also undecidable. Now do the conversion to ASCII code of (S3): (S7) 00000000h: 54 68 65 20 73 75 62 73 74 69 74 75 74 69 6F 6E ; The substitution 00000010h: 20 6F 66 20 91 78 92 20 62 79 20 68 69 73 20 6F ; of ‘x’ by his o 00000020h: 77 6E 20 71 75 6F 74 61 74 69 6F 6E 20 69 6E 20 ; wn quotation in 00000030h: 78 20 69 73 20 75 6E 70 72 6F 76 61 62 6C 65 ; x is unprovable Let be (S8) similar to (S4), replacing (S3) by (S7) (only the hexadecimal

numbers from the second to 17th columns). (S8) is also undecidable. This does not mean that there is an error in the theorem, nor results in an

impairment of their importance. Gödel’s result means the inability to automate the deduction of theorems if not severely restricts the classical language of arithmetic of Peano-Russell. Gaps that can be detected are in the logical system of Peano-Russell, not in Godel’s proof. He discovered a “black hole” in the logic-mathematical galaxy, which could not be avoided. And this is all that Gödel was committed himself at his time: to show the limitations of the Hilbert formalism, not a new property or relation of number theory.

*** A SUGESTION FOR FURTHER RESEARCH In the footnote 48a, p. 191 of the original memory, Gödel promises that in

Part II (which was never written) he would unveil the reasons of incompleteness, and advances specifically that “the undecidable statements presented here become decidable if we add appropriate higher types (e.g. type ω for P system).” For short: in 1931 apparently he thought there was a solution for incompleteness.

Since the demonstration of Rosser [1936], which expands to all consistent

and recursively axiomatizable extensions of PA system the undecidability, all attempts to find new axioms to avoid the incompleteness were abandoned.


In a supplementary note dated August 1963, and published in the edition of

van Heijenoort [1967] Gödel communicates the extension of its results also to any mechanical inference system.

Nothing he says about his change of perspective on the possibility of avoiding

the undecidable statements by adding higher types, or about the abandonment of the work plan for the promised Part II. It is understandably that, as say van Heijenoort [1967] in footnote 68a (from the publisher, not the author), Gödel has chosen not to develop the arguments outlined in Part I, for the ready acceptance of their results. But it is very strange not to give a detailed explanation about the “real reason for the incompleteness inherent to all formal systems,” which outlined in footnote 48a, and whose proof he promised right there. Stranger still is that nobody has asked for.

These statements (the original and the Rosser’s extensions) are undecidable

through the mechanism of self-substitution, and we have seen that it produces non-resolvable citations of nested functions and quotations with free variables. Call them “encrypted free variables”. We guess that this property: “being a formula which depends on citations with encrypted free variables” is recursive, so it is expressible in PA.

It is likely that the solution he had in mind would have made the PA system

in what is now called predicate calculus of higher order. That is, it would have exceeded the narrow limits of the order 1. But the axiom of complete induction was enunciated in his time with type 2 variables whose values are predicates (or sets if you will) and not individual numbers:

(P) { P(0) & (x) [ P(x) ⊃ P(x+1) ] } ⊃ (x) P(x) (For any property P, and any number x, if P holds for 0, and since it holds

for a number implies that holds for the next, then it holds for all numbers.) Today it is stated as an axiom scheme, i.e. a syntactic form common to

endless formulas, without speak about classes or properties, or quantify over predicates:

If Fml is a well formed formula , the following composite formula is an

axiom: { Fml(0) & (x)[ Fml(x) ⊃ Fml(x+1) ] } ⊃ (x) Fml(x)


The statements of the theorem VI (the original and Rosser’s extensions) are undecidable through the mechanism of self-substitution, and we have seen that it produces non-resolvable functions and nested citations with free variables. Call them “free variables encrypted”. The property of a formula to be dependent on citations with this kind of variables is recursive, i.e., is expressible in PA.

So, we would “emulate” an axiom with variables of “type ω” with the

following axiom scheme (thus keeping the system in order 1):

If A is a set of formulas exempt of encrypted free variables, and B is a formula that contains them, then any formula of the type “B is not deductible from A” is an axiom.

The axioms of PA and all its instances are exempt from free variables encrypted, and any extension that is obtained by adding formulas exempt of such variables to the original set of axioms, is covered by the scheme.

However, it is worth exploring the concept of formal system from the following viewpoint. The interest in “axioms” comes from the possibility of deriving formulas applicable to useful calculations in daily practice. Add as “axioms” certain formulas simply because they are not derivable nor is its negation, but we think they are “true” in any instance of interpretation, it goes against the ultimate purpose of the system. It is not advisable to take them as a hotbed of theorems. It is perfectly acceptable that there are four categories of arithmetic formulas: axioms, theorems, unprovable because they are negations of theorems, and blind spots of language, which are undecidable as some kind of open formulas (with free variables “in the transfinite”). If we can recognize them recursively, they will not produce any problems, nor do we need to pass judgment about its truth or falsity. Then, instead of being used as an axiomatic scheme recognition, the algorithm could rule the formulas that are well formed at a first level, but does not produce any fully extensional interpretation, in the sense of Herbrand.

With this categorization, the PA system would be, if not demonstrably

complete, but “potentially completable”. We cannot know a priori whether Goldbach's conjecture, for example, is met or not. But we can be sure that its eventual demonstration, or its denial, would not bring any contradiction. Nor it will automatically trigger another “Goldbach type conjecture” if we decided to include it as an axiom.

Floyd & Putnam [2000] argue a position that is a whole research program,

and imply a radical change in the perspective of logical-mathematical studies:


What if someone were to have said to Wittgenstein: “When I say that P is true in Russell's system, what I mean is simply that its translation into English -anyone of its mathematically equivalent translations, including ‘P is unprovable in PM’- is true?” We believe that Wittgenstein would have pointed out that the notion of truth is eliminable here. The highlighted sentence (emphasis added by us) could be considered a

central thesis in Wittgenstein’s philosophy of mathematics. When he says in [1956] Part VII, § 19: “My task is not to talk (e.g.) about Godel's theorem, but to by-pass it” , he is not trying to undermine the philosophical significance of that theorem, only that, given the normative nature which he attributed to mathematical propositions, the existence of truths beyond its demonstrability is an issue that transcends mathematics (and even metamathematics), and therefore, is a separable issue from the formulation of a set of calculation rules.

In short, and mimicking the wittgensteinian style: if someone tells me that he

has built a wonderful maze in the gardens of the campus, I have reason to worry because every turn can be a deadly trap.

The bad news is that the labyrinth exists and has no outlet. The good news is

that it may be not unavoidable. That is, from within the system we could detect when we have a tunnel without outlet in front of us, to avoid falling into it. Probably there remain unsolved mysteries about “how it is that the system allows that construction?” or “is it possible to build a system that avoids such statements?”. But after all, we could avoid confronting us with the unprovable mathematical truths.

The key to that future generations of scholars obtain some new result is an

attitude that pioneered Wittgenstein: prevent (rather than Platonic) the Pythagoreans reverently in front to a mathematical proof that encloses more than mathematics. The lucky future scholars must also thank Floyd and Putnam for to vindicate the Bemerkungen , and revive a debate that is not solved.

Abel-Luis Peralta Buenos Aires, December 2015.

Note 1.


The following are the equivalences used by Gödel in his original notation

symbols:

19 q(x,y) ≈ x ¬Bx [ Sb ( y ) ] Z(p) p(y) ≈ (x) q(x,y) r(x) ≈ q(x, Z(p) ) 17 Gen r ≈ (x) q(x, Z(p) ) Neg ( 17 Gen r ) ≈ ¬(x) q(x, Z(p) )

Replacing q:

19 17 Gen r ≈ (x) { x ¬Bx [ Sb ( p ) ] } Z(p)

Since "BX" is short for "Beweisbar" (demonstrable, in this case in the X class of

numerals of formulas), and the bar on the relation symbol means denial, and "Sb" stand for "Substitution", then using a similar Mendelson [1964] notation, it can be transcribed (in several steps) as follows:

Step 1) Let Fmlº be the open initial formula:

(x) ¬Proof (x, Subst (y, num (y), 219))

where: Proof (x, y) is a relationship that hold if x is a demonstration of y; Subst (y, num (y), 219) is an operation that replaces, in the numeral y the

numeric code 219 (corresponding to the variable 'y'), the numeral of the numeral (double numeral, even if seem strange).

Step 2) Let #Fmlº the numeric code of Fmlº .

Step 3) Replace both occurrences of the variable 'y' in Fmlº in step 1, by #Fmlº:

(x) ¬Proof (x, Subst (#Fmlº, num (#Fmlº), 219))


We propose call "auto-substitution" the operation in the 2nd argument (in the current literature is often used "diagonalization"), that given an open formula, replace the free variable with the code number of the whole formula that contains it.

***

Note 2. Solomon Feferman, in his paper of [1999], comments: Not only were

Gödel's result stunning, but also his own explanation of why they hold was surprising. This was given in a footnote that was apparently included in the paper[ 10] only as an afterthought, since it is numbered 48a. But it expressed a fundamental conviction of Gödel's which he reiterated throughout the rest of his life, and this conviction bring us close to the heart of our leading question. There is evidence that he thought such a view would be unacceptable to the Hilbert school, and that he must have hesitated to say anything of this sort at all. (Emphasis added by us).

This seems to imply that all his life thought that incompleteness, although

inherent in formal systems as we know them, perhaps with enough changes in the rules, and new axioms, it was solvable. But he did not disclose their attempts for avoid the questionings from Hilbert’s school. However, he worked more than 20 years, until his death, in the attempt to achieve a proof of the consistency of arithmetic, in the Hilbert's finitary terms. He published in 1958 a first article on the subject, in German, in the journal Dialectic, and in 1978 submitted to the same journal a translation by himself into English, a revised and expanded version, which was not published at the time, because the magazine folded, and he died. It was published in [1990]. He thought he had not been echoed in 1958 by being published in German.

He also believed that many unsolved problems of arithmetic could be

addressed with the addition of powerful and accurate enough axioms: “From the time of his stunning incompleteness results in 1931 to the end of his life, Gödel called for the pursuit of new axioms to settle undecided arithmetical problems.” … “While Godel's program to find new axioms to settle Continuum Hypothesis has not been realized, what about the origins of his program in the incompleteness results for number theory? As we saw, throughout his life G6del said we would need new, ever-stronger set-theoretical axioms to settle open arithmetical problems of even the simplest, purely universal, form-problems he called of Goldbach type.” Feferman [1999].


In these attempts he was totally isolated, probably because the scholars of the time viewed with distrust the Godel's “Platonism” , and also found that “solve” the arithmetic incompleteness ruin parallel with the indeterminism, established as paradigm in physics, as a corollary (legitimate or not, it's another matter) of the Heisenberg’s uncertainty principle, quantum mechanics and Einstein’s relativity theory. In fact, the academic activity focused on expanding to other fields and/or deepen mathematical undecidability and incompleteness, so the persistence of Gödel in some extent counteract the effects of their own negative results in the formal program, would be against the mainstream of the time, mainly mechanist and probabilistic; the ideas of Gödel after 1931 were a kind of anticlimax to that school.

If Jacques Herbrand would lived long enough, perhaps Gödel would had a

support and escort of his stature, with a more intuitionistic perspective, as we can guess from the Gödel’s [1934] recognition (see note 35) for their suggestions on the recursion theory , and what today is called Herbrand-Gödel computability (first paradigm of computability, predecessor and inspirer of Church and Turing, and probably much more rigorous than these).

Regarding the results of Turing, Church, and others, expanding the results of

his incompleteness theorems, Gödel says in a postscript of 1964 to the paper of [1934]: “Please note that the results mentioned in this postscript not set limits on the ability of human reason, but rather the possibilities of pure formalism in mathematics.”

*** Note 3: THE INDEFINABILITY OF AUTO APPLICATION Consider the following function of functions (see Kleene[1952], § 58): 1 si (Ek) φi(x)=k (φi is defined for x) θ(i, x) = 0 si ¬(Ek) φi(x)=k (φi is NOT defined for x) If we restrict ourselves to recursive functions, this function coincides with the

representative (also called characteristic) function of the provability relation (Bew, in Gödel [1931] definition number 46, pp. 182-186 Original), since all recursive functions are representable in the PA system.


A number theoretic function is primitive recursive if is set from a constant

function and the successor function, or other recursive functions, applying the rule of recursión:

φi(0) = p φi(x+1) = g(q, φi(x)) It can also be defined indirectly: φi(x) = µz[ f(x, z) = 0 ] ...where f (x, z) is primitive recursive, and μ symbolizes the minimum value

of z that satisfies the equation, if any. If not, simply the value of x considered is not set . In this case it is said to be a partial recursive function.

Obviously θ(i, x) is not recursive, because there is a decision procedure to

assign a value, if and only if φi functions are primitive recursive; if they are partial recursive not always we have a procedure to check if there exists a solution of the given equation. Then, without being exhaustive, we have the following scenarios:

1. φi(x) is primitive recursive, then θ (y, x) = 1. 2. φi(x) is partial recursive, and can prove that there is a solution to the

equation, then θ (i, x) = 1. 3. φi(x) is partial recursive, and can prove that there is no solution to the

equation, then θ (i, x) = 0. 4. φi(x) is partial recursive, and we cannot prove that there is a solution to the

equation, nor that there is not. Then we do not know what value θ(i, x) has, because if we would be able to test with all the numerical sequence, at some point we might find a solution (it is the elementary method when we have a μ operator) and in that case it would 1; or it could be that the operator μ never end, and in that case would be worth 0; but you could never tell what is the case.

It might seem that the four options above exhaust all the possibilities6.

However, consider a special function, and lets asume that whose numeral code (or Gödel number) is n:

6 From this point, we do not follow Kleene, it is a conjectural and personal development.


φn(x) = ∂(x) = θ(x, x) This definition implies that all functions should be defined in its own code,

because it is assumed that θ is always defined. This is consistent when the function does not depend on the value of another

function (even identical to itself). For example, if the function is 1 when the argument is even, and 0 when it is odd, it can be calculated without problems in their own numerical code, or when he is a constant (does not depend on its argument). But see what happens in the following case:

∂(n) = θ(n, n) (where n is the numeral code/Gödel number of ∂) Suppose that the result is 1. We see that depends on the value of another

instance of itself. But since there is no argument that can apply to the second instance, the code is that of an open formula. Then it should be independent of the variable, equal to 1 for all values of the function φn. (If implicitly assume that argument has its own code as an infinite regress would be generated, which would make it indefinite). But this would mean that if the function φn(n) is undefined θ(n, n) is equal to 1, contradicting the definition that assign 0 in this case. Absurd.

Suppose that the result is 0. In this case, as defined by θ, the function whose

code number is n should be indefinite in n. But n is just the code of ∂, i.e., φn; so it is undefined in n but at the same point has value 0. Absurd.

Therefore, the only consistent possibility is that is not defined in its own

code. The reason for this definition lack is that the argument of φn(n) is either an open code formula or indicator of an infinite sequence of nested functions (the procedure for constructing the argument never ends).

Thus interpreted, although the formula is correct from the standpoint of

formal language in a first level of analysis, it is not correct if we examine its argument, and this flaw in its construction have not consistent remedies.

Obviously, we present this not as solution, or a definitive result, but as a

suggested way to investigate the issue.


REFERENCES Bays, Timothy [2004]: “On Floyd and Putnam on Wittgenstein on Gödel”,

JPhil CI.4 (April 2004): 197–210. Bernays, Paul [1959]: “Comments on Ludwig Wittgenstein's Remarks on the

foundations of mathematics”. Ratio 2: 1-22. Berto, Francesco [2009]: “The Gödel Paradox and Wittgenstein's Reasons”

Philosophia Mathematica (2009) 17 (2): 208-219. Boolos, George; Burgess, John and Jeffrey, Richard [2007]: “Computability

and Logic”, Cambridge University Press, Fifth Ed. Chomsky, Noam[1957]: “Syntactic Structures”, Mouton & Co. Trad. esp.

“Estructuras sintácticas”, Siglo XXI. Davis, Martin [1965]: The Undecidable. Basic papers on undecidable

propositions, unsolvable problems and computable functions. Raven Press. Hewlett. New York.

Feferman, Solomon [1999]: Does Mathematics Need New Axioms? The

American Mathematical Monthly, Vol. 106, No. 2 (Feb., 1999), pp. 99-111. Floyd, Juliet and Putnam, Hilary [2008]: “Wittgenstein’s ‘Notorious’

Paragraph About the Gödel Theorem, Recent Discussions”, in Hilary Putnam: “Philosophy in an Age of Science: Physics, Mathematics and Skepticism” (Mario DeCaro and David Macarthur, eds. Harvard University Press, 2012).

[2000]: “A note on Wittgenstein's 'notorious paragraph' about the Gödel theorem”, The Journal of Philosophy, # 97, pp. 624-632.

Gödel, Kurt [1931]: “Über formal unentscheidbare Sätze der 'Principia

Mathematica' und verwandter Systeme I”, aus: Monatshefte für Mathematik und Physik, 38 (1931), 173-198.

English translation in van Heijenoort[1967]. Traducción española en Obras Completas, Ed. Jesús Mosterín[1981]. [1934]: “On undecidable propositions of formal mathematical systems”,

lecture notes by Stephen Cole Kleene and John Barkley Rosser (The Institute of


Advanced Study, Princeton, New Jersey); reprinted with corrections, emendations, and a postscript in Davis[1965].

Traducción española en Obras Completas, Ed. J. Mosterín. [1958]: “Über eine bisher noch nich benutzte Erweiterung des finiten

Standpunktes”, Dialectica, 12, págs. 280-287. [1980]: “On a hitherto unexploited extension of the finitary standpoint”,

Journal of Philosophical Logic 9 (2):133 - 142. [1981]: “Sobre una extensión de la matemática finitaria que todavía no ha

sido usada”. En Obras Completas, 1ª ed., editor y traductor Jesús Mosterín, Alianza Editorial, Madrid.

[1990]: “On an Extension of Finitary Mathematics which has not yet been Used”, (escrito en 1972), in Solomon Feferman, John Dawson & Stephen Kleene (eds.), Kurt Gödel: Collected Works Vol. II. Oxford University Press. 271--284.

Herbrand, Jacques [1930]: “Recherches sur la théorie de la démonstration”,

Comptes rendus, Académie des Sciences, Paris. English translation of Chapter 5, “The properties of true propositions”, in van Heijenoort[1967], pp. 525-581.

[1931]: “Sur la non-contradiction de l´arithmétique”, Journal für reine und angewandte Mathematik 166: 1–8, 1931 (con prólogo de Helmut Hasse), versión en inglés “The consistency of arithmetic”, en van Heijenoort[1967], pp. 618–628.

Hodges, Andrew [2014]: “Alan Turing: The Enigma”, Random House,

London. Kienzler, Wolfgang and Grève, Sebastian Sunday [2015]: “Wittgenstein on

Gödel: Reading Remarks on the Foundations of Mathematics, Part I, Appendix III, Carefully”, in “Wittgenstein and the Creativity of Language”, Palgrave Macmillan, 2015

Kleene, Stephen Cole [1964]: “Introduction to metamathematics”, North-

Holland Pub. Co. Kripke, Saul A.[1982]: “Wittgenstein on rules and private language”,

Blackwell Publishing Ltd., Oxford. Ladrière, Jean [1957]: “Les limitations internes des formalismes.” Louvain,

E. Nauwelaerts, Paris, Gauthier-Villars. Traducción española de José Blasco, Ed. Tecnos, Madrid, 1969.

Mendelson, Elliott[1964]: “Introduction to Mathematical Logic” (first

edition), D. Van Nostrand Company.


Quine, Williard Van Orman [1951]: “Mathematical Logic”, 2nd. ed., Harvard University Press. Trad. esp. Revista de Occidente, 1972.

[1970]: “Philosophy of Logic”, Prentice Hall. Trad. esp. Alianza, 1973. [1976]: “The Ways of Paradox and Other Essays”, Harvard University Press. Rodych, Victor [2003]: “Misunderstanding Gödel: New Arguments about

Wittgenstein and New Remarks by Wittgenstein”, Dialectica, Volume 57, Issue 3, pages 279–313, September 2003.

Rosser, John Barkley [1936]: “Extensions of some theorems of Gödel and

Church”, JSL, Vol. I, pp. 87-91. Russell, Bertrand [1905]: “On Denoting”, Mind, 14 : 479-493 Searle, John R. [1997]: “Roger Penrose, Kurt Gödel, and Cytoskeletons”, in

“The Mystery of Conciousness”, Granta Books. Shanker, Stuart G.[1988]: “Wittgenstein's Remarks on the Significance of

Gödel's Theorem”, in “Gödel’s Theorem in Focus”, Ed. Shanker, S.G., Routledge. Steiner, Mark [2001]: “Wittgenstein as his Own Worst Enemy: The Case of

Gödel's Theorem”, Philosophia Mathematica (2001) 9 (3): 257-279. van Heijenoort, Jean [1967]: “From Frege to Gödel”, Harvard University

Press. Wittgenstein, Ludwig [1956]: “Remarks on the Foundations of

Mathematics”, G.H. von Wright, R. Rhees, and G.E.M. Anscombe, eds.; Anscombe, trans. (Cambridge: MIT, 1978, revised edition), I, Appendix III, §8.

Yablo, Stephen [1993]: “Paradox without Self-Reference”, Analysis 53

(4):251-252.

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Bypassing Gödel

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